Moduli spaces of morse functions for persistence

Abstract

We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology and introduce new invariants to study these equivalence classes. These new invariants are as simple—but more discerning than—existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate any two Morse–Smale vector fields on the sphere by a sequence of fundamental moves by considering graph-equivalent Morse functions. We also explore the combinatorially rich world of height-equivalent Morse functions, considered as height functions of embedded spheres in \({\mathbb {R}}^3\). Their level set invariant, a poset generated by nested disks and annuli from level sets, gives insight into the moduli space of Morse functions sharing the same persistence barcode.

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Notes

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    The Morse–Smale complex described here is treated as a combinatorial structure, not to be confused with Morse–Smale–Witten chain complex (Banyaga and Hurtubise 2013, Chapter 7).

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Acknowledgements

This paper grew out of a productive discussion during the special workshop “Bridging Statistics and Sheaves” at the Institute for Mathematics and Applications in May 2018. The authors would like to thank the organizers for putting together the workshop, the IMA for hosting the event, and Mikael Vejdemo-Johansson for insightful conversations at the onset of this collaboration. The authors also thank the reviewers for helpful comments and suggestions. JC is partially funded by NSF CCF-1850052. JL is partially funded by EP/P025072/1. BTF is partially funded by NSF CCF-1618605 and DMS-1664858. BW is partially funded by NSF IIS-1513616 and IIS-1910733.

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Correspondence to Bei Wang.

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Catanzaro, M.J., Curry, J.M., Fasy, B.T. et al. Moduli spaces of morse functions for persistence. J Appl. and Comput. Topology 4, 353–385 (2020). https://doi.org/10.1007/s41468-020-00055-x

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Keywords

  • Persistent homology
  • Morse functions
  • Triangulations
  • Posets
  • Topological invariants

Mathematics Subject Classification

  • 58D29
  • 55N31
  • 37D15
  • 57M15
  • 05C22